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This classic experiment was first carried out by J.J.Thomson
in 1897. It involves the use of an electric field to accelerate
electrons up to high velocity, and a magnetic field to then steer the
electrons in a circular trajectory. The electrons are released by
thermionic emission from a heated filament (cathode) and are
accelerated towards a cylindrical anode. See Fig.1 and Fig.2. This is a similar
situation to the electron gun used to generate the electron beam in an
X-ray tube or in an electron microscope

Figure 1: Picture of the e/m apparatus.

Figure 2: Schematic of the e/m apparatus.

As the electrons accelerate from the cathode to the anode, they loose
potential energy, , and gain an equal amount of
kinetic energy, . Therefore they
arrive at the anode with a velocity given by:

(1)

Some of the electrons escape with this velocity though a narrow aperture
in the anode into a region where a uniform magnetic field exists. In
Fig. 1 this field, which is not shown would point out of of the paper
and towards the reader. The field exerts a magnetic force with a
magnitude of (since the velocity and the magnetic field are
perpendicular to each other). The direction of the force is
perpendicular to the velocity and acts as a centripetal force:

(2)

where is the radius of the path. Eliminating
between equations (1) and (2) one obtains the charge
to mass ratio:

(3)

The magnetic field is produced by the current flowing in two
Helmholtz coils. The distance between these circular coils is equal
to their radius, an arrangement that results in a reasonably uniform,
axial magnetic field midway between the coils. The magnitude of the
magnetic field, , in this region sis given by

We are going to determine the ratio by determining the
B-field and the radius of curvature of the electron beam in the magnetic
field. To do this you select an accelerating voltage and
keep it constant while you vary the magnetic field by varying the
current in the coils. For each magnetic field setting you will then
determine the radius of curvature of the beam. From each measurement
you should be able to extract the ratio and in the end
you will calculate the (weighted) average.

Select the following accelerating voltages:
150, 200 , 300 and 400 V. How well do you think you know the
voltage ?

For each voltage setting record the current in the coil
and the path diameter (or the radius) of the electron path. Make about
10 measurements equally spaced between the smallest and the largest current
for which you can still measure the path diameter. Estimate
the error of the diameter (radius)
measurement. How well to you know the current ?

Record your data for each voltage setting.

Measure the diameter of the Helmholtz coils and note the number of windings.

All measured quantities have an uncertainty associated with them as it
is impossible to measure any quantity with infinite precision. An
experimental uncertainty is also often called an experimental
error. This should NOT be confused with the difference between a
published or theoretical value and your experimental result. The
experimental uncertainty is often represented by the symbol . If
several experimental quantities contribute to your final result you
need to estimate how the uncertainties in your measurements will
affect the uncertainty in your final result. The simplest estimate is
to calculate the partial derivatives of your final results with
respect to each experimental quantity and add the various contributions
in quadrature.

Let’s assume you have measured the quantities
and , where are the
respective uncertainties (measurement errors). You use these
values to calculate a new quantity

(5)

You can now determine the uncertainty as follows:

(6)

Let’s apply this to the ratio . Note that all variables, :math:V, B, r` have
uncertainties .

, uncertainty in from reading the instrument

, uncertainty in this needs to be calculated since
the field is a function of and (see
(4) which are all
measured quantities that have uncertainties:

Assume you have several experimental values ,
where , for the same
quantity (e.g. ). You want to calculate the average of these
values taking into account the uncertainty associated with each
individual one. This
means that a measurement with a big uncertainty should not contribute
as much as a measurement with a much smaller uncertainty. This can be
done using the following expressions: